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Theorem hlbn 18780
Description: Every complex Hilbert space is a Banach space. (Contributed by Steve Rodriguez, 28-Apr-2007.)
Assertion
Ref Expression
hlbn  |-  ( W  e.  CHil  ->  W  e. Ban )

Proof of Theorem hlbn
StepHypRef Expression
1 ishl 18779 . 2  |-  ( W  e.  CHil  <->  ( W  e. Ban  /\  W  e.  CPreHil ) )
21simplbi 446 1  |-  ( W  e.  CHil  ->  W  e. Ban )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1684   CPreHilccph 18602  Bancbn 18755   CHilchl 18756
This theorem is referenced by:  hlcms  18783  hlprlem  18784
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-in 3159  df-hl 18759
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